Governing equations and boundary conditions

 
Figure 1:   Super-cavitating hydrofoil geometry.

A super-cavitating hydrofoil subject to a uniform inflow at an angle of incidence is considered, as shown in Figure 1. The ambient pressure is taken equal to . The chord length of the hydrofoil is taken equal to c and the length of the cavity is taken equal to l;SPMgt;c. Assuming that the flow is irrotational and incompressible, the total velocity vector, , may be expressed in terms of the perturbation potential, , as follows [1]:

The perturbation potential, , must satisfy Laplace's equation in the domain outside the cavity and hydrofoil:

 

In addition must satisfy the following conditions:

• Kinematic Boundary Condition: The total flow is required to be tangent to the wetted hydrofoil surface, as well as to the cavity surface. If is the unit normal to the hydrofoil or cavity surface directed into the fluid domain, then:

   

• Dynamic boundary condition on the cavity: The pressure is constant on the cavity surface and equal to (usually the vapor pressure). Defining the cavitation number, ,as follows:

  

  and applying Bernoulli's equation, the magnitude of total velocity on the cavity, , is found to be constant and given by

  The dynamic boundary condition then becomes:

   

  where is the unit vector tangent to the cavity surface.

• Kutta Condition: at the trailing edge of the foil or cavity (in the event of a super-cavity)
• Condition at infinity: at infinity

Equation 5 applies only in the case a cavity is present. In practice the cavitation number is known and the cavity length and cavity shape must be determined. Usually though the reverse problem is solved, in which case the cavity length is known and the corresponding cavitation number and cavity shape are determined.